The elliptic lambda is the fundamental modular function, taking values 0, 1, and infinity on the cusps.
The term “modular” comes from the moduli space of complex curves (a.k.a. Riemann surfaces) of genus 1 (Zagier, 1991).An example of a modular function is the j-invariant. In fact, every modular function is a rational function of the j-invariant (Snowden, 2020).
Formal Definition of a Modular Function
A modular function can be defined in several ways. One way is to view these as complex-valued functions on ℍ which are (Zagler, 1991):
- Invariant under the action τ ↦ (aτ + b)/(cτ + d) of Γ1 on ℍ,
- Holomorphic on ℍ,
- Satisfy suitable growth conditions at infinity.
A modular function can also be defined as one with the following properties:
- f(μ(z)) = f(z) for all μ in the modular group Γ.
- f has a Fourier expansion of the form
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Borwein, J. & Borwein, P. (1987). “Elliptic Modular Functions.” §4.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112-116.
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Stephanie Glen. "Modular Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/modular-function/
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